Variations of the Ramanujan polynomials and remarks on $\zeta(2j+1)/\pi^{2j+1}$

Abstract

We observe that five polynomial families have all of their roots on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang [MSW]. Our proofs rely upon theorems of Schinzel [S], and Lakatos and Losonczi [LL] and some generalizations.